As we have seen in our work with inequalities (see these entries to remind yourself if you need), an inequality states a range of solutions to a problem instead of just a singular answer.
The difference is best described with an example:
Here is the line $y=2x+3$y=2x+3
The line shows all the solutions to the equation. All the possible $y$y values that make this equation true for any $x$x value that is chosen.
For every $x$x value there is only one possible corresponding $y$y value.
For example, if $x=1$x=1, then according to the equation $y=5$y=5 (as marked on the diagram)
Here is the inequality $y>2x+3$y>2x+3
The solution to this is not a single line, as for every $x$x value, there are multiple $y$y values that satisfy the inequality. The solution graph is therefore a region.
A coloured in space indicating all the possible coordinates $\left(x,y\right)$(x,y) that satisfy the inequality.
For example, at $x=1$x=1, $y>5$y>5. So any coordinate with an $x$x value of $1$1 and a $y$y value larger than $5$5 is a solution.
The dotted line corresponds to the strictly greater than symbol that was used. That is, since $y$y cannot equal $2x+3$2x+3, we cannot include the points on the line.
Here is another example $y\le2x+3$y≤2x+3
Again we have a region, and this time we also have solid line indicating that the $y$y value can be less than or EQUAL to $2x+3$2x+3, for any given $x$x.
For example, if we choose $x=3$x=3, the points that satisfy the inequality are all the points whose $y$y value is less than or equal to $2\times3+3$2×3+3 or $9$9.
There are many points that do this. One such point would be $\left(3,8\right)$(3,8).
Examples
Question 1
Select the inequalities that describe the shaded region.
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A coordinate plane, with the x-axis from $-10$−10 to $10$10 and the y-axis also from $-10$−10 to $10$10. Two lines are drawn on the plane: a $solid$solid horizontal line which crosses the y-axis at $\left(0,-3\right)$(0,−3), and a $solid$solid line which crosses the x-axis at $\left(-\frac{5}{4},0\right)$(−54,0) and the y-axis at $\left(0,-5\right)$(0,−5). These lines intersect, dividing the coordinate plane into four regions. The $\text{upper left}$upper left region is shaded.
$y$y$\ge$≥$-4x-5$−4x−5 or $y$y$\ge$≥$-3$−3
A
$y$y$\ge$≥$-4x-5$−4x−5 and $y$y$\le$≤$-3$−3
B
$y$y$\le$≤$-4x-5$−4x−5 and $y$y$\ge$≥$-3$−3
C
$y$y$\le$≤$-4x-5$−4x−5 and $-\frac{5}{4}$−54$\le$≤$-3$−3
D
Question 2
Select the inequalities that describe the shaded region.
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A graph of lines of $y=x$y=x and $y=-x$y=−x on a cartesian plane is plotted. The line representing $y=x$y=x is illustrated as a dashed line. The line for $y=-x$y=−x is shown as a solid line. At the intersection of the two lines, the area to the left is shaded gray.
$y$y$>$>$x$x and $y$y$\le$≤$-x$−x
A
$y$y$\le$≤$x$x and $y$y$>$>$-x$−x
B
$y$y$>$>$x$xor$y$y$\ge$≥$-x$−x
C
$y$y$<$<$x$xand$y$y$\le$≤$-x$−x
D
Question 3
Outcomes
GM6-7
Use a co-ordinate plane or map to show points in common and areas contained by two or more loci
91033
Apply knowledge of geometric representations in solving problems
